Baccarat v10.0 Lucky Match

I am trying to find the house edge for this side bet .. 21st Century Baccarat v10.0

LUCKY MATCH PAY TABLE
WINNING HANDS PAYS
5 OR 6 OF A KIND 250 - 1
DOUBLE TRIPLETS 100 - 1
4 OF A KIND 30 - 1
FULL HOUSE 15 - 1
3 OF A KIND 6 - 1
2 OR 3 PAIRS 4 - 1

"The Lucky Match wager takes into account the cards in the banker’s hand and the cards in the player’s hand. If a player wagers on the Lucky Match Bonus Bet and the cards in banker’s hand in combination with the cards in the player’s hand make a qualifying hand as described below, the Lucky Match Bonus Bet wager shall win. For example, if the banker’s hand is composed of 2-K-4 and the player’s hand is composed of 4-K-4, then the Lucky Match Bonus Bet wager wins and is paid 15 to 1. A pair must be formed by two cards of the same rank, a Jack and a King can’t form a pair even though they have the same value of 0. Note that a 4-of-a-kind and a pair will be paid as a 4-of-a-kind."

I ran a 1 billion round sim and have a 8.87% HE.

Combinatorial analysis resulted in a 9.71% HE.

I can't post links/images due to my post count.

Any idea where I erred? Hoping someone out there has a solid number. Thanks in advance.

The analysis of this sidebet seems to me to be a bit tricky, because sometimes the banker and/or the player will stand on their first two cards. So you will never have "6 of a kind" of 4's, for example, and never have a full-house when both banker and player stand. Did you correctly take that into account?

Quote: gordonm888

The analysis of this sidebet seems to me to be a bit tricky, because sometimes the banker and/or the player will stand on their first two cards. So you will never have "6 of a kind" of 4's, for example, and never have a full-house when both banker and player stand. Did you correctly take that into account?
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Looking back at my scorecards I have 24 hands identified for four different shoes I’m looking at.

First one is seven hands of six cards pulled, nine hands of five cards pulled, and eight hands of four cards pulled for 24 consecutive hands.

The second one is six hands of six cards pulled, nine hands of five cards pulled and nine hands for four cards pulled for 24 consecutive hands.

The third shoe was eight hands of six cards pulled, twelve hands of five cards pulled and four hands of four cards pulled for 24 consecutive hands.

The fourth shoe was six hands of six cards pulled, five hands of five cards pulled and thirteen hands of four cards pulled for 24 consecutive hands.

Delete. Double posting.

Yes, I had to use the entire bacc universe of 302,500 possible 6 card hands to accomplish this. I first check each row for 4, 5 or 6 cards used and only evaluate those.

Here are my numbers for the sim and CA ..(I included a single pair just for checking)

5 or 6 of a Kind 71,873
Double Trips 123,004
4 of a Kind 2,454,014
Full House 8,010,490
3 of a Kind 41,583,707
2 or 3 Pairs 77,099,663
Pair 423,214,861
Other 447,442,388
1,000,000,000

5 or 6 of a Kind 330,651,336,960
Double Trips 580,991,385,600
4 of a Kind 11,880,028,677,120
Full House 38,888,044,339,200
3 of a Kind 208,563,763,445,760
2 or 3 Pairs 384,191,087,149,056
Pair 2,118,892,793,102,336
Other 2,235,070,916,067,328
4,998,398,275,503,360

Quote: SuperNatural

Yes, I had to use the entire bacc universe of 302,500 possible 6 card hands to accomplish this. I first check each row for 4, 5 or 6 cards used and only evaluate those.

Here are my numbers for the sim and CA ..(I included a single pair just for checking)

5 or 6 of a Kind 71,873
Double Trips 123,004
4 of a Kind 2,454,014
Full House 8,010,490
3 of a Kind 41,583,707
2 or 3 Pairs 77,099,663
Pair 423,214,861
Other 447,442,388
1,000,000,000

5 or 6 of a Kind 330,651,336,960
Double Trips 580,991,385,600
4 of a Kind 11,880,028,677,120
Full House 38,888,044,339,200
3 of a Kind 208,563,763,445,760
2 or 3 Pairs 384,191,087,149,056
Pair 2,118,892,793,102,336
Other 2,235,070,916,067,328
4,998,398,275,503,360
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Okay, I've taken a shot at calculating the number of combinations that are 4oak.

There are 4,998,398,275,503,360 total combinations.

4oak, TTTT-KKKK: 4,696,705,105,920 combinations

4oak, 2222- 9999: 7,577,256,453,120 combinations

Total 4oak: 12,273,961,559,040 combinations, 0.002455579


So, that's a slightly higher number than your combination analysis but is in good agreement with your simulation.

Quote: SuperNatural

Yes, I had to use the entire bacc universe of 302,500 possible 6 card hands to accomplish this. I first check each row for 4, 5 or 6 cards used and only evaluate those.
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SuperNatural,

The (10*9/2)^2*(10)^2 = 302,500 value applies only when all the 0-cards are lumped together, so we have only 10 ranks. If you have to treat all 13 ranks separately, then the number of combinations is (13*12/2)^2*(13)^2 = 1,028,196.

Hope this helps!

Dog Hand

I get slightly different numbers (though I haven't checked the logic just that they add up to the correct total), the T6 number does correspond to 9*32*31*30*39*28*27 (2 5 6 7 T J Q K A) and it does get your House Edge of 8,89%.

I'm so glad that help has arrived!

Charlie, the sum of your T42 and T4 numbers matches my combination math calculation for 4oak (posted earlier) to all 14 places! Which is a pretty good confirmatory spot-check, since 4oak straddles 4-card, 5-card and 6-card bacarrat hands.

So, I think we should assume CharliePatrick's results are most likely correct.

As it took a while to knock up a quick program I hadn't seen the "afternoon"'s updates when posting my results, but thanks if your figures agree with mine for 4oak; it gives me a warmer feeling.

Dog, turns out I was looping through all 13 ranks (1,399,489 not 1,028,196 combinations?) but my problem was I was not using a card value of 0 for 10,J,Q,K when passing them in as ranks to evaluate the bacc hand. This caused drawing errors so my 5 and 6 card hands weren't right.

All my numbers now agree with charliepatricks.

Thanks all!

I'd be happy to analyze this if someone can name a casino anywhere on earth that offers this side bet.

The Bicycle Casino in LA offers this sidebet, and their website even has the paytable Bicycle Casino

Lucky Match Sidebet documentation is also here: Lucky Match Sidebet Rules and Description

Quote: gordonm888

The Bicycle Casino in LA offers this sidebet, and their website even has the paytable Bicycle Casino

Lucky Match Sidebet documentation is also here: Lucky Match Sidebet Rules and Description
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Thank you. I'm on it. I see they have another side bet called the Super Tie. Please follow the link to see my analysis of that. I welcome a second opinion on it.

I agree with Charlie's numbers. That said, here is my new page on the Lucky Match. I welcome all comments.

You're gonna lose about 7 out of 8 bets, but hey the big payout is less than 300 to 1.

The probabilities of the rows on the paytable are interesting, particularly the ratios of probabilities of adjacent rows. I think it's unusual to have two adjacent rows on a payout table vary in probability by as much as 20X. Yet we have:

Double triplets 100 616,417,689,600 0.000123
Four of a kind 30 12,273,961,559,040 0.002456

The main reason is that Doublet triplets has restrictive requirements on teh ranks of all six cards, while 4oak only has restrictive requirements on 4 of 6 cards.

Another reason, though, is that double triplets requires a six card hand (both player and banker draw a card) and there are some combinations of ranks, such as (4,3) and (0,9) that are forbidden to be double triplets by the gameplay rules of Baccarat. Other combinations of ranks are limited in how they can be formed, which lower their probability. For example, a double triplet of (6,0) can be formed by:

P(0,0); D(6,6); 6, 0

but not by

P(0,6); D(0,0). X, X

IMO, its an interesting mathematical system.